This invention relates to a high activity supported catalytic component which is advantageously usable in the manufacture of highly crystalline .alpha.-olefin polymer and also to a method for homo- or co-polymerizing the .alpha.-olefin in the presence of the high activity supported catalytic component. More specifically stated, the high activity supported catalytic component is manufactured by including a liquid aromatic hydrocarbon and/or a liquid aromatic hydrocarbon halide as reforming agent in a catalytic component system consisting of a magnesium halide, a titanium compound and further an electron donor and, if required, further by using a halogen concentration diluent therewith; and in the method, the .alpha.-olefin is homo- or co-polymerized by using a catalytic system in which the high activity supported catalytic component is concomitant with an organoaluminum compound and an electron donor.